2
votes
1answer
129 views

Is RCA-Rudin one of the worst textbooks? [closed]

Someone told me that "Real and complex analysis - rudin" is actually rated a bad textbook among researchers, since it gives no motivation. Is it true? I agree that this text provides less ...
2
votes
0answers
50 views

Is it generally difficult to memorize 'multivariable calculus' theorems?

There are many weak forms of "Fubini's theorem" with strong hypotheses in elementary calculus texts. However, these strong hypotheses are very unnatural and are thus hard to memorize. Compared to ...
3
votes
1answer
66 views

Recommendation on studying differential geometry

Below are what i studied so far: Rudin - Principles of Anlysis (only except one chapter, namely differential forms) Munkres - Topology (only point-set topology) Rudin - RCA (Only first 4 ...
16
votes
11answers
788 views

Why limits work

I'm currently a first year student in electrical engineering and computer science. I know how to compute limits, derivatives, integrals with respect to one variable that is things from one variable ...
0
votes
0answers
24 views

List of crucial results deserving more attention for first course in Real Analysis

Can it help to form a list of crucial results for basic courses that are concealed as exercises or neglected? I don't know of other resources for this, as I wrote here. I am happy for this to be ...
-5
votes
2answers
152 views

(soft) How hard REALLY is Advanced Calculus I and Abstract Algebra I [closed]

WTF everyone here downvote Arthur Fischer he is asking for it!!! This next fall, I will be taking Advanced Calculus I and Abstract Algebra I, along with Physical Chemistry I, Computer Science I, ...
3
votes
3answers
113 views

How to know if I'm doing real analysis correctly?

I've just started a second year course in real analysis. This is my first proof-oriented course. Last year, our maths curriculum was introductory tertiary calculus and algebra. When I practised ...
1
vote
1answer
22 views

Reference request to study Borel summation

Could someone recommend sources to learn about Borel summation procedure? Books, articles or reviews? I have a background in basic analysis.
1
vote
1answer
40 views

Can Bohr-Mollerup Theorem be extended to complex plane?

There is a famous theorem says about Gamma function: Bohr-Mollerup Theorem Let $f:(0,\infty)\rightarrow \mathbb{R}^+$ be a function satisfying below (i) $f(x+1)=xf(x), \forall x\in ...
5
votes
2answers
90 views

Intuition for differentiating beneath the integral

I apologize in advance for a vague question. There is a theorem: If both $f(x,s)$ and $\partial _sf(x,s)$ are continuous in $x$ and $s$, then $$\partial_s\int_a^bf(x,s)\,dx=\int_a^b ...
2
votes
0answers
86 views

recommendation on studying real-analysis

I hope someone who had studied RCA would answer my question. I bought Rudin-Real and Complex analysis for self-study. (Please don't mind that it is a self-study. I don't care at all whether a text is ...
2
votes
1answer
88 views

Derivative existence theorem

Has anyone here heard of the Derivative existence theorem? Derivative existence theorem: For $f$ defined on some interval including $a$, $f$ is differentiable at $a$ if and only if there ...
1
vote
2answers
114 views

Infinite sum with 0 terms: comparison to infinite product

Depend on what text you read, an infinite product with an infinite number of terms that are 0 is either divergent, or diverge to 0. Even though, obviously, the partial product is still a convergence ...
10
votes
1answer
210 views

“Why” is $[\mathbb{C}:\mathbb{R}] < \infty$?

Obviously this question is a little open-ended. A lot of complex analysis seems to work primarily because we can view $\mathbb{C}$ as a finite-dimensional $\mathbb{R}$-algebra, and apply analytic and ...
20
votes
3answers
726 views

Who came up with the $\varepsilon$-$\delta$ definitions and the axioms in Real Analysis?

I've seen a lot of definitions of things like boundary points, accumulation points, continuity, etc, and axioms of the real numbers. But I have a hard time accepting these as "true" definitions or ...
3
votes
1answer
210 views

Real Analysis and Statistics

What level of real analysis do you think is desirable for the study of statistics? I know that for many statisticians with applied focus, rigorous mathematics tend to give them a headache and I am ...
3
votes
2answers
179 views

How does one get better at real analysis proofs?

How does one proceed through a math proof in real analysis? My instructor always says make a diagram, but I am not a visual learner. It seems that whenever I write out the definition of an assumption, ...
1
vote
0answers
47 views

A Question Related to the Divergence Test for Series

Suppose $\lbrace x_i\rbrace$ is a sequence of real numbers. If $\lim x_i=0$ then for every $\varepsilon>0$ there exists an $N$ such that $$n\ge N\Rightarrow \vert ...
0
votes
0answers
49 views

Between any rational and $\sqrt{2}$ is another rational [duplicate]

I'm self-studying analysis from Rudin, and in there is this proof. I understand the proof after reading it, but how would a mathematician come up with this or approach doing a proof like this? It ...
1
vote
1answer
75 views

Analysis for Engineers: Where Do You Start?

Having taken none of the prerequisite rigorous treatments of mathematics during my undergrad years, I feel at a disadvantage to the people in my major what do have that analysis/abstract math ...
2
votes
2answers
81 views

Why study difference equation, sequences, etc?

A couple of weeks ago I studied a thorough(yet basic) course in ODE, part of an course in analysis. Shortly after, we moved on to the study of difference equations, which is very much similar to ...
1
vote
2answers
107 views

Why is the undergraph definition of Lebesgue integral so rare?

So in Pugh's Real Mathematical Analysis, the initial definition of the Lebesgue integral is as the Lebesgue measure of the undergraph of the function (where the function is nonnegative, with the usual ...
5
votes
2answers
194 views

Is $(-\infty,\infty)$ a closed **interval**?

Note that we are working in the reals, not the extended reals. Do you understand a closed interval as "an interval that is a closed set" or as "an interval that includes both its endpoints"? If the ...
1
vote
1answer
57 views

Is a closed n-dimensional disk compact necessarily compact?

As the title asks, is a closed n-dimensional disk compact necessarily compact? I'm thinking the answer would be no. If you consider the case in $\mathbb{R}^1$ then can you define the radius to be ...
1
vote
0answers
75 views

On Cauchy Sequences

I would consider this a soft question because I am seeking some insight on how to work with Cauchy sequences by using the Cauchy criterion for convergence. To my understanding, the definition is ...
2
votes
5answers
149 views

What's so special about $e$? [duplicate]

If someone with not much mathematics in his luggage asks me: What is so special about $\pi$? then off course I have an answer. Even if $i$ would be the subject (I allready see him gazing at my ...
0
votes
0answers
45 views

Exercise references

I could recommend any good text analysis, or perhaps a list of exercises with good problems (for show) on dips, submersiones and implicit functions. I appreciate any references.
2
votes
0answers
105 views

Is my general approach to proofs acceptable? A general topology example.

Proving: $A$ is closed iff $A = \bar{A}$. "To the right": If $A$ is closed, $ A = \bar A$ If $A$ is closed this means that it contains all of its own accumulation points. And we would find that its ...
4
votes
1answer
493 views

Unexpected Practical Applications of Calculus

Calculus shows up in a lot of places in the world. Specifically, here are three areas where I see it used the most: Optimization problems. Anything involving rates of change (e.g. velocity ...
3
votes
2answers
98 views

difference between $\mathbb{R}^2$ and $\mathbb{R} \times \mathbb{R}$

I was going through some of notes in regards to Fourier analysis and I noticed that in some cases when dealing with a 2 dimensional transform the function $f \in \mathbb{R}^2$ while other times $f \in ...
1
vote
0answers
85 views

Real and Rational Numbers

Intuitively, we often think of real numbers as existing in one-to-one correspondence with the points on a continuously drawn line, the real number line. One way of expressing the completeness of the ...
4
votes
2answers
95 views

Any finite set is a null-set

How can we prove that a finite set is a null-set? Maybe would it be easier to prove that the outer measure of a finite set is $0$? any ideas on how to tackle this problem? thanks,
7
votes
1answer
269 views

How to write well in analysis (calculus)?

This is kind of a subjective question, I know; often I find myself failing exams and homeworks because of the way i write down proofs. Either I don't know how to start, or somehow the main point of ...
15
votes
6answers
731 views

What are the main uses of Convex Functions?

Up till now I have just learned that the concept of convexity in functions of one variable is used to complete the graphs of functions, meaning to locate points of inflexion and see if the graph is ...
1
vote
2answers
274 views

Pure mathematics for engineers

I have recently completed my first year of Eng. Physics taking the standard math courses: Calculus, Linear Algebra 1 and 2, Multivariable Calculus and Numerical Analysis. Recently though I have been ...
2
votes
4answers
382 views

Advanced undergraduate(?) Real Analysis book which is concise and lots of interesting problems

I have gone through the other book recommendations on Real Analysis, but I think my requirements and background is slightly different. I am a Physics undergrad teaching myself Pure math. My journey is ...
5
votes
3answers
857 views

Baby Rudin vs. Abbott

I am considering Stephen Abbott's Understanding Analysis and Walter Rudin's Principles of Mathematical Analysis. I am looking for a comparison between the two that addresses both of the following ...
5
votes
2answers
110 views

State-of-art of the Discrete Fourier Transform

I would like to know what is the state-of-art in the research of the discrete Fourier transform. I have listed some questions to help answering, please add your own to make the list more ...
6
votes
1answer
162 views

Is there an integral form of Newton's method?

Warning : This seems like a silly sort of question, not the kind I'd ask out loud. The contraction mapping theorem is a basic tool for proving existence of, and finding solutions to, equations. Given ...
1
vote
1answer
184 views

Which topics of real-analysis should be studied if you have already done calculus

Which parts of real-analysis are worth studying if you have already taken several calculus courses? I know that real-analysis is more 'rigurous', but still I wonder whether it is worth to again go ...
4
votes
2answers
1k views

What are the real-world applications of real analysis?

I've read the wikipedia article on mathematical analysis and this, but I can't exactly find an answer. Is real analysis just some pure math, or does it really have something to with physical ...
4
votes
4answers
224 views

Is there a symbol for the idea of the smallest value greater than zero?

I know that it isn't actually a number but I do think it's a concept in mathematics. So the question is, is there a symbol representing this concept? I thought maybe it was Phi but I couldn't find it ...
6
votes
3answers
135 views

Analogy between Integration and Summation

There are many analogies between definite integral and Summation: $$\int_a^b \leftrightarrow \sum_a^b$$, This makes me wonder if there is analogous concept of indefinite integral, derivative and ...
0
votes
1answer
80 views

Similar textbook to Konigsberger's Analysis 2?

I am currently taking an introductory course to real analysis and my professor has decided to leave Rudin's "Principles of Mathematical Analysis" when teaching us the concepts of Lebesgue integration. ...
3
votes
0answers
243 views

Is the $ϵ,δ$ definition of a limit not well-defined?

I just watched this youtube video: http://www.youtube.com/watch?v=K4eAyn-oK4M He lays out his objections against the $ϵ,δ$ definition around 14 min. Here is the discription of the video: In ...
8
votes
2answers
446 views

How to deal with Homeomorphisms?

I have one doubt that may be too general, I don't know, so sorry if this is not a good place to ask it. I've also seem many other people with the same problem that I have, so I think that if this ...
2
votes
1answer
300 views

What kinds of sets are reasonable to place on the continuum?

Warning: I don't know anything about set theory so I wouldn't really know how to spot an existing answer if it were around. Suppose I want to model some economic good or product. I would like to ...
13
votes
1answer
423 views

Real analysis textbok that develops the subject in a self-motivated, coherent fashion?

Well, it seems as though I just failed my analysis prelim for the second time... I have one more try in about $5$ months. I'm failing to build up a framework for how to think about analysis problems. ...
18
votes
1answer
313 views

Learning Aid for Basic Theorems of Topological Vector Spaces in Functional Analysis

I am self-teaching myself the basics of functional analysis (e.g. topological vector spaces), and frankly I am starting to get a migraine sorting out/organizing in my head all of the ...
1
vote
2answers
98 views

What should I know about Sobolev space?

I have done some Sobolev spaces with some embedding theorems, trace theorems etc. Sorry that my question is really vague. If my professor asks me what is great about Sobolev space, what should I ...